OPTIMIZATION PROBLEMS

IntroductionIn class, we started encountering the idea of absolute maximums and absolute minimums. These are important concepts! We often want to find the best of something given some constraints. Let’s try some problems.

Background: Many people are involved in deciding how to package the products you see in grocery stores. A company will want the strongest container using the least amount of material possible. Packaging engineers select materials and package shapes to adequately protect the product through shipping at a reasonable cost. A container’s shape, as well as its material, is important in determining its strength. From an engineering perspective, the sphere is the strongest form, followed by the circular cylinder. The rectangular box comes in a poor third. An infinite number of dimensions can be used to construct a right circular container of a given volume or a rectangular box of a given volume. What would packaging engineers need to consider?

Situation: You have just been hired as a packaging engineer for a company. You are to evaluate the current packaging for one canned good and one boxed good to determine if they are the optimal dimensions (minimize materials and maximize volume). For shipping purposes and design purposes, the company insists that the depth of the boxed good is two inches. Your supervisor expects a report on your findings, including the calculations to support your claim. You want to make a good impression, so you decide to create a model of one of the goods in the optimal form and you persuade a friend in marketing to help with the design.